Frame related sequences in tensor product of Hilbert spaces

Abstract

We study frames and related sequences in tensor products of separable Hilbert spaces from an algebraic perspective. For sequences $\{f_n{\}}_{n\in \mathbb{N}}\subset H_1$ and $\{g_m{\}}_{m\in \mathbb{N}}\subset H_2$, we consider the tensor sequence $\{f_n\otimes g_m{\}}_{n,m\in \mathbb{N}}$ in $H_1\otimes H_2$. We characterize the lower semi-frame and Riesz--Fischer properties of tensor sequences in terms of the corresponding properties of the component sequences. These results extend to infinite tensor products. In addition, we study the action of tensor sums of operators on frame-related sequences in tensor product Hilbert spaces.